A novel splitting strategy to accelerate solving generalized eigenvalue problem from Kohn--Sham density functional theory

TL;DR

A novel eigenpair-splitting method, inspired by the divide-and-conquer strategy, for solving the generalized eigenvalue problem arising from the Kohn-Sham equation, which focuses on solving a series of subequations defined on the entire domain.

Abstract

In this paper, we propose a novel eigenpair-splitting method, inspired by the divide-and-conquer strategy, for solving the generalized eigenvalue problem arising from the Kohn-Sham equation. Unlike the commonly used domain decomposition approach in divide-and-conquer, which solves the problem on a series of subdomains, our eigenpair-splitting method focuses on solving a series of subequations defined on the entire domain. This method is realized through the integration of two key techniques: a multi-mesh technique for generating approximate spaces for the subequations, and a soft-locking technique that allows for the independent solution of eigenpairs. Numerical experiments show that the proposed eigenpair-splitting method can dramatically enhance simulation efficiency, and its potential towards practical applications is also demonstrated well through an example of the HOMO-LUMO gap calculation. Furthermore, the optimal strategy for grouping eigenpairs is discussed, and the possible improvements to the proposed method are also outlined.

Figures (12)

• Figure 1: Three states for the hydrogen atom.
• Figure 2: First: Mesh on the root tetrahedron. Seond: Mesh on the global refinement of the tetrahedron. Third and forth: the local refinements of second mesh.
• Figure 3: The octree data structure.
• Figure 4: Twin-tetrahedron and four-tetrahedron.
• Figure 5: Flowchart of the multi-mesh adaptive algorithm for the KS equation.